Basic Network Creation Games
MohammadTaghi HajiAghayi
AT&T Labs & U. of Maryland
Joint work with
Noga Alon, Erik Demaine, and Tom Leighton
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Motivations for Network Creation Games
Distributed way to create a network
 Undirected graph.
 Nodes  Selfish agents
 Agents’ cost:

 Creation cost (network design)
 Usage cost (network routing)
Combine two costs by defining the cost of an
edge to be α and agents minimize their sum
 Several studies so far: FLMPS03, CP05,
AEEMR06,DHMZ07,HM07,A08,DHMZ09,etc

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Basic Network Creation Games
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The simplest and the heart of all such games while
avoiding α
Motivations: Cash-oblivious model
No cost transformation but every edge cost is the
same
Thus each edge only perform edge swap, replacing
an existing edge with another incident edge
We focus on structures of equilibria
 Diameter
 Price of anarchy

(PoA)
Agents try to minimize sum or maximum
distance to all other vertices
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Formal Definitions

Sum equilibrium: for every edge vw and every
node w’, swapping edge vw with vw’ does not
decrease the total sum distances from v to all
others
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Max equilibrium: for every edge vw and every node
w’, swapping edge vw with vw’ does not decrease
the max distances from v to all others
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Our Results

For sum equilibrium:
bound 2 O(√ log n) for diameter (and thus PoA) and
lower bound 3 for general graphs
 Give an evidence (and a conjecture) for a polylog upper
bound for diameter (and PoA)
 Tight bound 2 for trees
 Upper

For max equilibrium:
bound √n for diameter in general graphs even for
insertion-stable equilibria
 Tight bound 3 for trees
 Lower
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Sum Equilibrium on Trees: Diameter 2
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Swap for v is a net win unless sb+sw<=sa
Swap for w is a net win unless sv+sa<=sb
Summing up: sb+sw+sv+sa<=sa+sb
Contradiction since sw+sv>=2 by definition
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Max Equilibrium on Trees: Diameter 3
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None of three swaps around a with edge av is
helpful for the local diameter of a
Proof of upper bound is different from Sum
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A Diameter-3 Sum Equilibrium Graph
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The proof is involved and uses some lemmas and
case analysis
We do not know any example of diameter 4 or higher!
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A Diameter- √n Max Equilibrium Graph
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The proof is cute and more involved
The graph is indeed insertion-stable as well
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Sum Equilibrium: Diameter 2 O(√ log n)
First we need the following lemma whose
proof is a little bit involved:
 Lm: In any sum equilibrium, the addition of
any edge uv decreases the sum of distance
from u by at most 5n lg n
 Bk(u) denote the number of vertices within
distance at most k from u and Bk= minu Bk(u)
 We prove either B4k>n/2 or B4k >=(k/20lg n)Bk
 Assume there is a u with B4k(u)<= n/2.
Certainly B3k(u)<= n/2.
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Diameter
Ai
O(
logn)
2 √
Ci
(Cont.)
3k
ti
v
D  N3k (u)  N3k 1 (u)
D
u
3k
≤3k+1
t1
C1 : dist  2k
>2k
w
l
C2
D   Ci
i 1
• Distance of any v outside of B3k(u) from one vertex of T is at most dist(u,v)- k
• By Pigeonhole principle, there are at least n/(2|T|) vertices Ai whose distances from
the same vertex t in T is at most dist(u,v)- k
• Adding an edge from u to t improves the sum of distances from u by at least
(k-1) n/(2|T|) <= 5nlg n (by the Lm) which implies |T|>= k/(20lg n)
•The balls of radius k centered at the vertices of T are all pairwise disjoint, all lie within
distance 4k from u and each of them has at least Bk vertices. Thus B4k >=(k/20lg n)Bk
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Evidence for polylog diameter
An n-vertex graph is ε-distance uniform if there is a
value r such that from every vertex v the number of
vertices w at distance exactly r from v >= (1- ε)n
 We can connect high-diameter sum equilibria graphs
to high-diameter distance uniformity. More formally:
Thm: Any sum equil. graph G with at least 24 vertices
and diameter d> 2lg n induces an ε-distance–uniform
graph G’ with n vertices and diameter (εd/lg n )
 Conj: Distance-uniform graphs have diameter O(lg n)
 If Conj above is correct, Thm gives O(lg2n) diameter
 We know Conj is true e.g., for Cayley graphs
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Main Open Problem
Can we prove a polylog upper bound (even
O(log n) or smaller) for diameter of sum
equilibria esp. by proving the conjecture?
 Consider convergences of (basic) network
creation games
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